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Limits to the muon flux from WIMP annihilation in the center of the Earth with the

AMANDA detector

J. Ahrens,1 E. Andres,2 X. Bai,3 G. Barouch,4 S.W. Barwick,5 R.C. Bay,6 T. Becka,1 K.-H. Becker,1 D. Bertrand,7

A. Biron,8 O. Botner,9 A. Bouchta,8, ∗ S. Carius,10 A. Chen,4 D. Chirkin,6, 11 J. Conrad,9 J. Cooley,4

C.G.S. Costa,7 D.F. Cowen,12 E. Dalberg,2, † C. De Clercq,13 T. DeYoung,4, ‡ P. Desiati,8 J.-P. Dewulf,7

P. Doksus,4 J. Edsjo,2 P. Ekstrom,2 T. Feser,1 T.K. Gaisser,3 M. Gaug,8, § L. Gerhardt,5 A. Goldschmidt,14

A. Goobar,2 A. Hallgren,9 F. Halzen,4 K. Hanson,12 R. Hardtke,4 T. Hauschildt,8 M. Hellwig,1 G.C. Hill,4

P.O. Hulth,2 S. Hundertmark,5 J. Jacobsen,14 A. Karle,4 J. Kim,5 B. Koci,4 L. Kopke,1 M. Kowalski,8

J.I. Lamoureux,14 H. Leich,8 M. Leuthold,8 P. Lindahl,10 P. Loaiza,9 D.M. Lowder,6, ¶ J. Ludvig,14 J. Madsen,4

P. Marciniewski,9, ∗∗ H.S. Matis,14 C.P. McParland,8 T.C. Miller,3, †† Y. Minaeva,2 P. Miocinovic,6

P.C. Mock,5, ‡‡ R. Morse,4 T. Neunhoffer,1 P. Niessen,13 D.R. Nygren,14 H. Ogelman,4 Ph. Olbrechts,13

C. Perez de los Heros,9, §§ A. Pohl,10 R. Porrata,5, ¶¶ P.B. Price,6 G.T. Przybylski,14 K. Rawlins,4

W. Rhode,11 M. Ribordy,8 S. Richter,4 J. Rodrıguez Martino,2 P. Romenesko,4 D. Ross,5 H.-G. Sander,1

T. Schmidt,8 D. Schneider,4 E. Schneider,5 R. Schwarz,4 A. Silvestri,11, 8 M. Solarz,6 G.M. Spiczak,15

C. Spiering,8 D. Steele,4 P. Steffen,8 R.G. Stokstad,14 O. Streicher,8 P. Sudhoff,8 K.H. Sulanke,8 I. Taboada,12

L. Thollander,2 T. Thon,8 S. Tilav,3 M. Vander Donckt,7 C. Walck,2 C. Weinheimer,1 C.H. Wiebusch,8, ∗

C. Wiedemann,2 R. Wischnewski,8 H. Wissing,8 K. Woschnagg,6 W. Wu,5 G. Yodh,5 and S. Young5

(The AMANDA collaboration)1Institute of Physics, University of Mainz, D-55099 Mainz, Germany

2Department of Physics, SCFAB, Stockholm University, S-10691 Stockholm, Sweden3Bartol Research Institute, University of Delaware, Newark, DE 19716, USA4Department of Physics, University of Wisconsin-Madison, WI 53706, USA

5Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA6Department of Physics, University of California, Berkeley, CA 94720, USA

7Universite Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium8DESY-Zeuthen, D-15735 Zeuthen, Germany

9Division of High Energy Physics, Uppsala University. S-75121 Uppsala, Sweden10Department of Technology, Kalmar University, S-39182 Kalmar, Sweden11Fachbereich 8 Physik, BUGH Wuppertal, D-42097 Wuppertal, Germany

12Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA13Vrije Universiteit, Dienst ELEM, B-1050 Brussels, Belgium

14Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA15Department of Physics, University of Wisconsin-River Falls, WI 54022, USA

(Dated: February 1, 2008)

A search for nearly vertical up-going muon-neutrinos from neutralino annihilations in the centerof the Earth has been performed with the AMANDA-B10 neutrino detector. The data collectedin 130.1 days of live-time in 1997, ∼109 events, have been analyzed for this search. No excessover the expected atmospheric neutrino background has been observed. An upper limit at 90%confidence level has been obtained on the annihilation rate of neutralinos in the center of the Earth,as well as the corresponding muon flux limit, both as a function of the neutralino mass in the range100 GeV-5000 GeV.

PACS numbers: 95.35.+d, 95.30.Cq, 11.30.Pb

I. INTRODUCTION

There are strong observational indications for the ex-istence of dark matter in the universe. Measurementsof the energy density of the universe, Ω0, from the com-bined analysis of cosmic microwave background radia-tion data and high red-shift Type Ia supernovae favorΩ0 = 1, with a matter, ΩM, and a cosmological con-stant, ΩΛ, component. Combined with data from rota-tion curves of galaxies and cluster mass measurements,the matter contribution to Ω0 is 0.3 ≤ ΩM ≤ 0.4. BigBang nucleosynthesis calculations of primordial helium,

lithium and deuterium production, supported by abun-dance measurements of these elements, set an upper limiton the amount of baryonic matter that can exist in theuniverse, ΩB ≤ 0.05 (see Ref. 1 for a recent review ofvalues of Ω). Non-baryonic dark matter must thereforeconstitute a substantial fraction of ΩM.

In this paper we present results of a search for non-baryonic dark matter in the form of weakly interactingmassive particles (WIMP) using the AMANDA high-energy neutrino detector. The next section contains abrief motivation for WIMPs as dark matter candidates.Section III describes briefly the characteristics of the

2

AMANDA detector in the configuration used for thisanalysis. Sections IV and V contain a description of thesimulation and analysis techniques used. In section VI wediscuss the sources of the current systematic uncertain-ties of our analysis. In section VII we present the resultsof the analysis and we introduce a novel way of calculat-ing upper limits in the presence of systematic uncertain-ties. An upper limit on the neutrino-induced muon fluxexpected from WIMP annihilation in the center of theEarth is derived with this method. A comparison withpublished muon-flux limits obtained by existing neutrinoexperiments is presented in section VIII.

II. WIMPS AS DARK MATTER CANDIDATES

Particle physics provides an interesting dark mat-ter candidate as a Weakly Interacting Massive Parti-cle (WIMP). The relic density of particle type i de-pends on its annihilation cross section, σ, as Ωih

2 ∼3 × 10−27/ 〈σv〉 (neglecting mass-dependent logarithmiccorrections), where 〈〉 indicates thermal average and vis the relative velocity of the particles involved in thecollision (see for example Ref. 2). Weak interactions pro-vide the right annihilation cross section for the WIMPsto decouple in the early universe and give a relic densitywithin the required range to contribute substantially tothe energy density of the universe today. This is basicallywhat would be needed to solve the dark matter problem.

In particular, and starting from a completely differentrationale, the Minimal Supersymmetric extension to theStandard Model of particle physics (MSSM) provides apromising WIMP candidate in the neutralino, χ. Theneutralino is a linear combination of the B-ino, B, andthe W-ino, W, the supersymmetric partners of the elec-troweak gauge bosons, and of the H0

1 and H02, the neutral

Higgs bosons, and it is stable (assuming R-parity con-servation, which is further supported to avoid too rapidproton decay). The actual composition of the neutralinocan have cosmological consequences since its annihilationcross section depends on it. For example, it has beenargued that a mainly W-ino type neutralino would notbe cosmologically relevant in the present epoch since itwould have annihilated too fast in the early universe toleave any relevant relic density3.

Still, the large parameter space of minimal supersym-metry can be exploited to build realistic models whichprovide relic neutralino densities within the cosmologi-cally interesting region of 0.025<

∼ Ωχh2 <1. Negative re-sults from searches for supersymmetry at the LEP accel-erator at CERN have set a lower limit on the neutralinomass mχ > 31 GeV (Ref. 4), while theoretical argumentsbased on the requirement of unitarity set an upper limitof 340 TeV (Ref. 3). Imposing in addition the condi-tion on Ωχh2 mentioned above, only models with mχ

<∼

10 TeV (Ref. 5) become cosmologically interesting.Neutralinos have a non-negligible probability of scat-

tering off nuclei of ordinary matter. Assuming the dark

matter in the Galactic halo is (at least partially) com-posed of relic neutralinos, elastic interactions of theseparticles with nuclei in the Earth can lead to energylosses that bring the neutralino below the escape veloc-ity, becoming gravitationally trapped6,7. For high neu-tralino masses (> a few hundred GeV) direct capturefrom the halo population by the Earth is kinematicallysuppressed8. In this case neutralinos can be accretedfrom the population already captured by the solar sys-tem. Gravitational capture is expected to result in an ac-cumulation of neutralinos around the core of the Earth,where they will annihilate. An equilibrium density isreached when the capture rate equals the annihilationrate. Neutrinos are produced in the decays of the result-ing particles, with an energy spectrum extending overa wide range of values and bounded from above by theneutralino mass. Annihilation of neutralinos directly intoneutrinos (or light fermion pairs in general) is suppressedby a factor m2

f /m2χ due to helicity constraints, where mf

is the fermion mass.Neutrino detectors can therefore be used to constrain

the parameter space of supersymmetry by setting limitson the flux of neutrinos from the center of the Earth2,9.Note that this indirect neutralino detection will be fa-vored for high neutralino masses, since the cross sectionof the resulting neutrinos with ordinary matter scaleswith Eν .

III. THE AMANDA-B10 DETECTOR

The AMANDA-B10 detector consists of an array of302 optical modules deployed in ten vertical strings atdepths between 1500 m and 2000 m in the South Pole icecap. The strings are arranged in two concentric circlesof 60 m and 120 m diameter respectively. The moduleson the four inner strings are separated by 20 m in thevertical direction, while in the outer six strings the ver-tical separation between modules is 10 m. An opticalmodule consists of a photomultiplier tube housed in aspherical glass pressure vessel. Coaxial cables (in the in-ner 4 strings) and twisted quad cables (in the outer 6strings) provide the high voltage to the photomultipliertubes and transmit the signals to the data acquisitionelectronics at the surface.

Muons from charged-current high-energy neutrino in-teractions near the array are detected by the Cherenkovlight they produce when traversing the ice. The rela-tive timing of the Cherenkov photons reaching the opti-cal modules allows the reconstruction of the muon track.A more detailed description of the detector is given inRef. 10. The detector was triggered when a majority re-quirement was satisfied: an event was recorded if at least16 modules had a signal within a predefined time windowof 2µs. The data taking rate was 100 Hz.

AMANDA-B10 was in operation during the 1997Antarctic winter. The separation of 300 atmosphericneutrinos from the data sample collected in that period

3

established the detector as a high-energy neutrino tele-scope11. The array was upgraded with 122 more modulesduring the antarctic summer 1997/1998 and in 1999/2000253 additional ones were added, completing the proposeddesign of 677 optical modules in 19 strings, AMANDA-II12.

IV. SIGNAL AND BACKGROUND

SIMULATIONS

A. Simulation of neutralino annihilations

Neutralinos can annihilate pair-wise to, e.g., ℓ+ℓ−, qq,W+W−, Z0Z0, H0

1,2H03, Z0H0

1,2 and W±H∓. Neutrinosare produced in the decays of these annihilation products.Neutrinos produced in quark jets (from e.g. bb or Higgsbosons) typically have lower energy than those producedfrom decays of τ leptons and gauge bosons. We will referto the first type of annihilation channels as “soft” and tothe second as “hard”.

The simulations of the expected neutralino signal weredone in the framework of the SUSY models described inRef 13. The hadronization and decay of the annihilationproducts have been simulated using PYTHIA14. The sim-ulations were performed for six different WIMP massesbetween 10 GeV and 5000 GeV. For each mass, six dif-ferent annihilation channels (cc, bb, tt, τ+τ−, W+W−

and Z0Z0) were considered, with 1.25×106 events gener-ated for each. Note that the decay of b– and c–hadronswill take place in matter instead of vacuum. This wasincorporated in the simulations in an effective mannerjustified by the fact that, for the neutralino masses con-sidered, the re-interactions of these heavy hadrons withthe surrounding medium are not dominant, and can beparametrized as an effective energy loss at the time ofdecay. As a reference soft spectrum, we chose the anni-hilation into bb, and as a reference hard spectrum, theannihilation into W+W−. For a given mass, these twospectra can be regarded as extreme cases. We have usedthese channels in the analysis described below, bearingin mind that a typical spectrum would lie somewhere inbetween.

B. Simulation of the atmospheric neutrino flux

Neutrinos from the decay of secondaries produced incosmic ray interactions in the atmosphere constitute thephysical background to the neutralino search. We havesimulated this atmospheric neutrino flux using the cal-culations of Lipari15. To obtain the rate of neutrino in-teractions producing muons we have used the neutrinoand anti-neutrino−nucleon cross sections from Gandhi et

al.16. The actual neutrino-nucleon interactions have been

simulated with PYTHIA using the CTEQ317 parametriza-tion of the nucleon structure functions. The use ofPYTHIA allows to model the hadronic shower produced

at the vertex of the interaction and, therefore, to calcu-late the Cherenkov light produced by secondaries. Whenthe neutrino-nucleon interaction occurs within the instru-mented volume of the detector, this is a non-negligiblecontribution to the total event light output.

A three-year equivalent atmospheric neutrino samplewith energies between 10 GeV and 10 TeV and zenithangles between 90 (horizontal) and 180 (verticallyup-going) has been simulated18. The sample contains3.7×107 events, of which 41234 triggered the detector.

C. Simulation of the atmospheric muon flux

The majority of the triggers in AMANDA are inducedby muons produced in cosmic ray interactions in the at-mosphere and reaching the detector depth. The simula-tion of this atmospheric muon flux was performed usingthe BASIEV19 program. We note that this program onlyuses protons as primaries. However, the systematic un-certainty introduced by this approximation is negligiblein comparison with that from the present uncertaintyin the primary flux intensity. Moreover, heavier nuclearprimaries produce more muons per interaction, but withlower energies on average20, which will in general looseall their energy and decay before reaching the detector.A study performed using the CORSIKA21 air shower gen-erator, with the QGSJET option to model the hadronicinteractions, including the complete cosmic ray composi-tion confirms this scenario.

The simulation of a statistically significant sampleof atmospheric muon background is an extremely highCPU-time consuming task due to the strong rejectionfactors needed. We have simulated 6.3 × 1010 primaryinteractions, distributed isotropically with zenith angles,Θ, between 0 and 85 degrees, and with energies, E, be-tween 1.3 TeV and 1000 TeV, assuming a differential en-ergy distribution ∝ E−2.7(Ref. 22). The total number oftriggers produced were 5 × 106. Normalizing to the pri-mary cosmic ray rate, the generated sample correspondsto about 0.6 days of equivalent detector live-time. Due tothe narrow vertical angular cones used for this analysisthis background sample is sufficient to model the detec-tor response and develop the rejection cuts. In addition,a larger sample of background data was used in the train-ing of the discriminant analysis program used as cut level4. This is described in more detail in the next section.

D. Muon propagation

The muons produced in the signal and backgroundsimulations described above were propagated from theproduction point to the detector taking into account en-ergy losses by bremsstrahlung, pair production, photo-nuclear interactions and δ-ray production from Ref. 23.The Cherenkov light emitted by the secondaries produced

4

cosΘ

even

ts

atmospheric muon MCData

1

10

10 2

10 3

10 4

10 5

10 6

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

FIG. 1: Angular distributions of data and atmospheric muonsimulation Monte Carlo (MC) at different analysis levels. Topto bottom: trigger to level 4. The distributions are normalizedto the simulated sample, 5×106 events.

in these processes is taken into account when calculatingthe response of the detector to the passage of the muon.

V. DATA ANALYSIS

The analysis presented in this paper was performedon data taken with the 10-string AMANDA detector be-tween March and November 1997. The experimentaldata set consists of 1.05×109 events in a total of 130.1days of detector live-time. The data were first cleaned ofnoise hits and hits from optical modules that were un-stable during the running period. Short pulses, that arelikely induced by cross talk between channels are alsorejected at this stage. Details on the data cleaning pro-cedure are given in Ref. 24. The data is then recon-structed and five filters consisting of cuts based on theevent hit pattern and the quality of the reconstructionare applied in order to identify potential up-going neu-trino candidates. The distributions of the reconstructedzenith angle from trigger level (after hit cleaning) untilfilter level 4 for data and simulated atmospheric muonsare shown in figure 1. The curves have been normalizedto the simulated sample, 5×106 events. The uppermostcurves in the plot show the reconstructed direction with-out any quality criteria applied to the fits, showing goodagreement between the data and the Monte Carlo sam-ple along the whole angular range. The curves clearlyindicate that a small percentage (about 2%) of the origi-nally down-going tracks are misreconstructed as up-going

cosΘ

arb.

uni

ts

atmospheric neutrino MCmχ=100 GeV, χχ→bb

–

mχ=5000 GeV, χχ→W+W-

10-4

10-3

10-2

10-1

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2

FIG. 2: Angular distribution of muons from atmospheric neu-trinos and from the annihilation of neutralinos after filter level2. The two extreme neutralino masses and annihilation chan-nels considered in this paper are shown. The relative normal-ization is arbitrary.

(cos Θ less than zero the figure). The series of cuts de-scribed below were developed to reject such misrecon-structions, and their effect on the angular distribution isalso shown in figure 1 for comparison. The filter level2 and level 3 curves show that the filtering procedure ismore effective rejecting the simulated muon backgroundthan the data. This is due to detector effects not includedin the simulation of the detector response and survivingto these levels, like electronic cross talk between channelsor inefficiencies of the digitizing electronics. Other pro-cesses not included in the background simulations thatcan contribute to the discrepancy are overlapping eventsfrom uncorrelated cosmic ray interactions and the con-tribution from electron neutrino induced cascades. Toaccount for this different behavior between data and sim-ulated background under standard cuts, we have used aniterative discriminant analysis as cut level 4 (see subsec-tion VD) trained on a sub-sample of data (which rep-resents the real remaining background better than thesimulations) and a sub-sample of the neutralino signal.A final series of high quality cuts were applied after thediscriminant analysis, bringing the remaining data sam-ple to agree with the number of events expected fromthe known atmospheric neutrino flux, as shown in fig-ure 4 and table I. Note that the atmospheric neutrinocurve and the data curve in figure 4 join and follow eachother in the last two steps of the cuts applied within thelevel 5 filter. The next subsections give a more detaileddescription of the variables used and the cuts applied at

5

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103

Neutralino mass (GeV)

Eff

icie

ncy

Level 1, hard spectrum

Level 1, soft spectrum

Level 2, hard spectrum

Level 2, soft spectrum

FIG. 3: Efficiencies relative to trigger level at filter levels 1and 2 as a function of the neutralino mass.

each filter level.

A. Filter level 1

In a first stage, a simple and computationally fast filterbased on fitting a line to the time pattern of the events,was applied to the data sample in order to reject obvi-ous down-going tracks. This “line fit” (LF) assumes thatthe known space point of each hit optical module, ~ri, isrelated to the measured hit time, ti, by ~ri = ~ro + ~vti.The minimization of χ2 =

∑

i(~ri − ~ro −~vti)2, where the

index runs over all the hits in the event, leads to an ex-plicit solution for ~v. The zenith angle of the fitted trackis readily obtained as cosΘLF = −vz/|v|. The angularresolution of the line fit is relatively low since it doesnot incorporate any information about the geometry ofthe Cherenkov cone or about scattering of the Cherenkovphotons in the ice. Still, its simplicity and computationalspeed makes it a very useful tool for a first assessmentof the track direction and for rejection of down-going at-mospheric muons25. The first level filter rejected obviousdown-going atmospheric muons by requiring ΘLF > 50.

B. Filter level 2

The events that pass the level 1 filter are reconstructedusing a maximum likelihood approach (ML) as describedin10. In short, the ML technique uses an iterative processto maximize the product of the probabilities that the op-tical modules receive a signal at the measured times, with

1

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FIG. 4: Rejection and efficiency at each filter level for thedata and simulations of the neutralino signal, atmosphericneutrinos and atmospheric muons. The dashed part corre-sponds to rejection levels surpassing the statistical precisionof the simulated sample, yielding zero remaining events. Theneutralino signal curve should be read only with respect tothe right axis scale, and it shows the relative signal efficiencywith respect to trigger level. The rest of the curves are plottedwith respect to the left axis scale.

the track direction (zenith and azimuth angles) as freeparameters. The expected time probability distributionsinclude the scattering and absorption characteristics ofthe ice as well as the distance and relative orientation ofthe optical module with respect to the track26.

The level 2 filter consists of two cuts: the ML-reconstructed zenith angle must be larger than 80 andat least three hits must be “direct”. A hit is defined asdirect if the time residual, tres (the difference between themeasured time and the expected time assuming the pho-ton was emitted from the reconstructed track and didnot suffer any scattering), is small. Unscattered pho-tons preserve the timing information. Therefore, the re-construction of the direction of tracks with several di-rect hits presents a significantly better angular resolu-tion. The number of direct hits associated with a track isthe first quality requirement applied to the reconstructeddata and simulated samples24. A residual time intervalbetween -10 ns and 25 ns was used to classify a hit asdirect at this level.

Figure 2 shows the zenith angle distributions of sim-ulated muon tracks from neutrinos produced in annihi-lation of neutralinos for the two extreme masses usedin this analysis as compared to that from atmosphericneutrinos after filter level 2. The corresponding curvefor data and simulated atmospheric muons is included in

6

figure 1. The combined effect of these two filters on thedata is a rejection of 98%, as shown in table I. The effi-ciencies with respect to trigger level of both level 1 andlevel 2 filters for simulated neutralino signal are shownin figure 3, for different neutralino masses and the twoextreme annihilation channels used.

Filters 1 and 2 are applied in an initial data reductioncommon to the different subsequent analyses of the data.The rest of the cuts described below were specificallydesigned for the WIMP search with the aim of identifyingand rejecting misreconstructions while maximizing signaldetection efficiency and background rejection27.

C. Filter level 3

The angular distribution of the events is the most ob-vious difference between the predicted neutralino signaland both the atmospheric neutrino flux and the atmo-spheric muon background. Neutrinos from neutralinoannihilations in the center of the Earth would be con-centrated in a narrow cone close to the vertical direction,while atmospheric neutrinos are distributed isotropically.The level 3 filter further restricted the ML-reconstructedzenith angle to be larger than 140, placed a cut on thetotal number of hit modules in the event, Nch > 10, andon the summed hit probability of the modules with a sig-nal, Phit >0.23. The number of hits with time residualsbetween -10 ns and 25 ns was required to be larger than4 and the number of hits with residuals between -15 nsand 75 ns to be larger than 5. At this stage the possiblecorrelations between the variables are ignored, and thecuts applied to each of them individually. Table I showsthe efficiency and rejection power at this cut level. Only5×10−4 of the simulated atmospheric muon backgroundsurvive this level, compared with 68% of the simulatedneutralino signal and 10% of the atmospheric neutrinos.

D. Filter level 4: Iterative Discriminant Analysis

To account for possible correlations between thevariables and perform a multidimensional cut inparameter space, the next filter level was basedon an iterative non-linear discriminant analysis, us-ing the IDA program28. Given a set of n vari-ables, the program builds the “event vector” x

k =(x1, ..., xn, x2

1, x1x2, ..., x1xn, x22, x2x3, ..., x

2n), where xi is

the value of variable i in event k. A class of events, thesignal or background sample, is characterized by theirmean vector 〈xs〉 or 〈xb〉, and the mean difference be-tween the samples is given by the vector ∆µ = 〈xs〉−〈xb〉.The spread of the variables is contained in the vari-ance vectors, µk

s = xk − 〈xs〉 and µk

b = xk − 〈xb〉,

which are used to define a variance matrix for each class,

V s,b =∑Nevts

k µks,b(µ

ks,b)

T, where Nevts is the num-ber of events in the signal or background samples and T

denotes the transpose. The problem of separating sig-

nal from background is transformed into the problem offinding a hyperplane in event vector space which givesminimum local variance for each class and maximum sep-aration between classes. This is translated into the re-quirement that the ratio R=(aT∆µ)2/aT V a should bemaximal, where here the variance matrix V is the sum ofthe variance matrices for signal and background and a isa vector of coefficients to be determined by training theprogram on a signal and a background sample. A targetsignal efficiency and background rejection factor are cho-sen beforehand. The coefficients a are determined in aniterative process carried out until the specified rejectionfactor is achieved or a predefined number of iterationsreached. The coefficients found in this way are used toselect events from the signal region in the multidimen-sional parameter space: each event is characterized bythe scalar D=a

Tx and a cut on D serves as the selection

criterion.

Eight variables were used in the training of the dis-criminant analysis program and in the subsequent cuts:the velocity of the line fit, the number of direct hits, thenumber of modules hit, the number of modules hit in thestring with the largest number of hits, the number of de-tector layers with a hit1, the extension of the event alongthe three coordinate axes, the average hit probability andthe probability that the event time pattern is compatiblewith that expected from a vertical up-going muon. Thisset of variables includes combined information from thefit track parameters as well as the general spatial andtemporal topology of the event.

Since to a first approximation the data consist of at-mospheric muon background, seven days of data, evenlydistributed along the year, were used as the backgroundtraining sample. For the signal training sample, muonsfrom the simulations of 250 GeV neutralinos annihilatinginto a hard spectrum were used. The combination of arelatively low neutralino mass and annihilation into thehard channel was chosen as giving a “typical” muon spec-trum. The target signal efficiency was set to 98% per iter-ation and the target global background rejection to 1000.The stopping criterion was set to 9 iterations, based onthe fact that further loops would reduce the number ofevents in the training sample to a too low number to berepresentative of the whole data set. The rejection ofbackground achieved was 220 with respect to cut level 3since the nine loops were exhausted before reaching thedesired rejection. The overall signal efficiency attainableafter the training process is then (0.98)9=0.83. The effectof the discriminant analysis event selection is shown intable I. It indeed achieves the expected signal efficiency,retaining 82% of the signal with respect to the previ-ous cut level. The discrepancy of the expected numberof atmospheric neutrinos and the number of remainingdata events at this level indicates that the data sam-

1 The detector was divided in eight horizontal layers of 65 m depth.

7

TABLE I: Rejection of data, of the simulated atmospheric neutrinos and atmospheric-muon background samples and efficiencyfor the simulated neutralino signal from trigger level to filter level 5.

Filter Level Data Atmospheric neutrinos Atmospheric muons χχ → WW

130.1 days 130.1 days equivalent 0.6 day equivalent mχ=250 GeV(events) (events) (events) (% of trigger level)

0 1.05×109 4899 5×106 1001+2 2.3 × 107 2606 7×104 793 1.2 ×106 472 2588 684 5441 89 13 565 14 16.0 0 29

ple is still contaminated by poorly reconstructed down-going muons. A last cut level was therefore developed toimprove the rejection of the remaining misreconstructedevents and select the truly up-going tracks.

E. Filter level 5: Final event selection

The remaining events after the discriminant analysiswith a zenith angle larger than 165 were passed throughthe following series of cuts. The length spanned by thedirect hits when projected along the track direction wasrequired to be at least 110 m, and the vertical lengthcontaining all hits was required to be at least 170 m. Thez component of the center of gravity of the direct hits(zc.o.g. =

∑

izi/Ndirect hits, where the sum is over all the

direct hits in the event) was required to be deeper than1590 m, and the percentage of hits in the lower half ofthe detector less than 55%. These cuts reject events witha spatially uneven concentration of hits, typically due todown-going atmospheric muons that pass just outside thedetector or stop close to the array.

The remaining data at this level are consistent withthe expected atmospheric neutrino flux. Figure 5 showsthe angular distribution of the remaining 14 data eventsand the remaining 16.0 simulated atmospheric neutrinoevents. The angular range shown is for Θ >165, the re-gion where a possible neutralino signal is expected to beconcentrated. No statistically significant discrepanciesare found between the expected number of events andangular distributions of the atmospheric neutrino back-ground and the data. This result is also consistent withthe results on atmospheric neutrinos presented in Ref. 11.

Due to the different angular shapes of the neutralinosignal for different neutralino masses (see figure 6 for thetwo extreme cases considered), we have chosen to restrictfurther in angle the signal region we use to extract thelimit on an excess muon flux. We use angular cones thatcontain 90% of the signal for a given neutralino mass.The remaining data and simulated atmospheric neutrinobackground events for the different angular cones usedare shown in table II. The background rejection powerand signal efficiency from filter level 1 to 5 are shown infigure 4 along with the effect on the data sample.

VI. SYSTEMATIC UNCERTAINTIES

An essential quantity when deriving limits, as we doin the next section, is the effective volume, Veff, of thedetector. It is the measure of the efficiency to a givensignal and it is defined as

Veff =nL5

ngen

Vgen, (1)

where nL5 is the number of signal events after filter level5 and ngen the number of events simulated in a volumeVgen surrounding the detector. The effective volume ofAMANDA-B10 as a function of muon energy is shownin figure 7. Given a MSSM model producing a muonflux with a given muon energy spectrum, the effectivevolume of the detector for this particular signal is alsocalculated through equation 1. This is shown in figure 8for the different neutralino masses used in this analysis.The shaded bands in both figures indicate the systematicuncertainty estimated as described below.

The evaluation of Veff is subject to experimental andtheoretical systematic uncertainties present in the anal-ysis. We have performed a detailed study of the effect ofthe uncertainty in several variables on the resulting ef-fective volume by propagating variations in any of themto the final evaluation of Veff.

Measurements of the scattering and absorptionlengths, λs and λa, using pulsed and DC light sourcesdeployed with the detector at different depths and YAGlaser light sent from the surface through optical fibers,have shown that these quantities exhibit a depth de-pendence which is correlated with dust concentration atdifferent levels in the ice29. A simulation of the detec-tor response including layers of ice with different opticalproperties has been developed and used to evaluate itseffect on the results. The effects introduced are muon-energy dependent and therefore dependent on the neu-tralino model. The effective volumes calculated with thelayered ice model are reduced between 1% and 20% withrespect to the uniform ice model, except for the lowerneutralino mass and soft annihilation channel (100 GeV)where the effect reaches 50%.

A further correction accounts for the uncertainties inthe optical modules’ total and angular sensitivities. It

8

cosΘ

even

ts

0

1

2

3

4

5

6

7

8

9

10

-1 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97

FIG. 5: Angular distribution of the remaining data events(dots) and simulated atmospheric neutrino events (shadedarea) at filter level 5. The angular range shown is between165 and 180. The shaded area represents the total un-certainty in the expected number of events.

cosΘ

% o

f ev

ents

mχ=5000 GeV (hard spectrum)

mχ=100 GeV (soft spectrum)

10-3

10-2

10-1

-1 -0.995 -0.99 -0.985 -0.98 -0.975 -0.97

FIG. 6: Angular distribution of the remaining fraction ofneutralinos at filter level5 with respect to trigger level fromthe two extreme neutralino masses studied in this paper.The angular range shown is between 165 and 180.

is known that during the process of re-freezing after de-ployment, air bubbles appear in the column of ice thathas been melted, changing locally the scattering lengthof the ice and distorting the effective optical module an-gular sensitivity with respect to that measured in thelaboratory. We have used a specific ice model for theice in the holes that accommodates this effect. The factthat it appears after deployment and that it is not di-rectly measurable in the laboratory makes it difficult toassess. Only by an iterative process of comparison ofdata and different hole-ice models can it be quantified.We estimate this effect to yield and increase of 20% ineffective volume with respect to the uniform angular re-sponse model with, again, the soft annihilation channelof the lowest mass neutralino giving a stronger effect of34%. An additional 20% uncertainty on the total opticalmodule sensitivity has been used.

The way to combine of all these effects into a final esti-mate of the total uncertainty in Veff is a difficult subject,since they are not independent contributions. As de-scribed in the previous paragraphs, by varying the initialparameters used in the simulations of the detector andin the ice properties, we have obtained a range of possi-ble values for the effective volume, which we consider asequally probable giving our current understanding of thedetector. We have chosen to take the nominal Veff to beused in equation 1 as the middle value of this range. Asa conservative estimate of the uncertainty we take halfthe width of the range of values obtained. We thus con-clude that our current estimate of Veff is affected by a

systematic uncertainty σVeff/Veff between 10% and 25%

depending on the neutralino mass considered, the lowermass of 100 GeV giving the larger relative error. A sim-ilar estimate including the same effects has been madefor the atmospheric neutrino Monte Carlo. In this casewe estimate the uncertainty on the effective volume foratmospheric neutrinos to be 20%.

Further uncertainty in the number of expected atmo-spheric neutrinos (column 3 in table I) is caused bythe uncertainties present in the calculation of the atmo-spheric neutrino flux. This is estimated to be of the orderof 30% in the energy region relevant to this analysis, andoriginates mainly from uncertainties in the normalizationof the primary cosmic ray spectrum and in the hadroniccross sections involved30. This has been taken into ac-count as an additional effect on top of the experimentaluncertainty on the effective volume for atmospheric neu-trinos, as described in section VII B.

It has recently been shown that different muon prop-agation codes can produce differences in the muon fluxand energy spectrum at the detector depth (see for ex-ample Ref. 31). The code used in this analysis uses theLohmann23 parametrizations for muon energy loss, whichproduce results in agreement within about 10% of morerecent codes32 for muon energies up to a few of TeV.We have not included any systematics arising from thetreatment of muon propagation in the ice in this analysis.

9

TABLE II: Number of data events, simulated atmosphericneutrino background events and the corresponding N90 forthe angular cones containing 90% of the signal for the dif-ferent neutralino masses. These angular cuts are applied inaddition to the level 5 filter described in section V. “s” and“h” denote the soft and hard annihilation channels. The num-bers in parenthesis in column 5 show N90 obtained withoutincluding systematic uncertainties.

mχ Angular cut Data Atmospheric N90

(GeV) (deg) (evts.) neutrinos (evts.)100s 167.5 10 12.1 9.2(4.7)100h 168.5 9 10.8 6.6(4.7)250s 170.0 7 8.6 5.9(4.1)250h

500s172.0 5 6.1 5.6(3.9)

1000s 173.0 4 4.6 5.3(3.9)500h 173.5 4 4.6 5.3(3.9)1000h

3000s174.0 4 3.9 5.6(4.7)

3000h

5000s5000h

174.5 3 3.9 4.4(3.6)

VII. RESULTS

From the observed number of events, nobs, and thenumber of expected atmospheric neutrino backgroundevents, nB, an upper limit on the signal, Nβ , at a cho-sen confidence level β%, can be obtained. We have usedthe unified approach for confidence belt construction33

to calculate 90% confidence level limits. In section B be-low we briefly describe a novel way of calculating limitsin the presence of systematic uncertainties that we haveused to obtain the final numbers presented in this paper.

A. Flux limits: the standard approach

For detectors with a fixed geometrical area A, it isnatural to derive a muon flux limit directly through φµ ≤Nβ/A · t, where t is the detector live-time. However, dueto the large volume of AMANDA and the lack of sharpgeometrical boundaries it is the effective volume Veff, asdefined in equation 1, that has to be used to determinea limit on the volumetric neutrino-to-muon conversionrate, Γνµ. The effective volume provides a measure ofthe detector efficiency since, in addition to through-goingtracks, it takes into account the effect of tracks startingor stopping within the detector. A limit can then be seton Γνµ, that is, on the number of muons with an energyabove the detector threshold Ethr produced by neutrinointeractions per unit volume and time,

Γνµ ≤N90

Veff · t(2)

Γνµ includes all the detector threshold effects and

model dependencies, as indicated below, and can be di-rectly related to a more physically meaningful quantity,the annihilation rate, ΓA, of neutralinos in the center ofthe Earth through

Γνµ(mχ) = ΓA ·1

4πR2⊕

∫ mχ

0

∑

Bχχ→X

(

dNν

dEν

)

× σν+N→µ+...

(Eν |Eµ ≥ Ethr) ρN dEν ,

(3)

where the term inside the integral takes into account theproduction of muons through the neutrino-nucleon crosssection, σν+N , weighted by the different branching ratiosof the χχ annihilation process and the corresponding neu-trino energy spectra, Bχχ→X dNν/dEν . ρN is the nucleondensity of the ice and R⊕ is the radius of the Earth. Wehave used a muon energy threshold of 10 GeV in the sim-ulations of the signal, which has been taken into accountthrough the muon production cross section.

Equation 3 is solved for ΓA. ΓA depends on the MSSMmodel assumptions, as well as the galactic halo modelused, being related to the capture rate of neutralinos inthe Earth. Different neutralino models predict differentcapture and annihilation rates that can be probed by ex-perimental limits set on ΓA. The right column of table IIIshows the limits thus derived for ΓA. The correspond-ing curves are shown in figure 9. Quoting limits on theannihilation rate has the advantage that the detector ef-ficiency and threshold are included through equation 2and, therefore, numbers published by different experi-ments are directly comparable. This is not usually thecase when presenting limits on muon fluxes, where atleast the detector energy threshold enters in a non triv-ial way and prevents direct comparison between experi-ments. However, since it is common in the literature topresent limits on the muon flux per unit area and time,we transform below our limit on ΓA into a limit on themuon flux from neutralino annihilations in the center ofthe Earth.

The total number of muons per unit area and timeabove any energy threshold Ethr within a cone of halfangle θc as a function of the annihilation rate is

φµ(Eµ ≥ Ethr, θ ≥ θc) =ΓA

4πR2⊕

∫ ∞

Ethr

dEµ

∫ π

θc

dθd2Nµ

dEµdθ,

(4)

where the term d2Nµ/dEµdθ represents the number ofmuons per unit angle and energy produced from the neu-tralino annihilations, and includes all the MSSM modeldependencies for neutrino production from neutralino an-nihilation and the neutrino-nucleon interaction kinemat-ics, as well as muon energy losses from the productionpoint to the detector. The upper limits on the annihila-tion rate are thus converted to a limit on the neutralino-induced muon flux at any depth and above any chosenenergy threshold and angular aperture. The 90% confi-dence level upper limits on the annihilation rate and the

10

Eµ (GeV)

Vef

f (m

3 )

10 3

10 4

10 5

10 6

10 7

10 8

102

103

FIG. 7: Effective volume of the detector as a function ofmuon energy at filter level 5.

10 4

10 5

10 6

10 7

10 8

102

103

104

Hard spectrum

Soft spectrum

Neutralino mass (GeV)

Vef

f (m

3 )

FIG. 8: Effective volumes for the neutralino signal as afunction of the neutralino mass.

TABLE III: The 90% confidence level upper limits on the muon flux from neutralino annihilations in the center of the Earth,φµ, for a muon energy threshold ≥ 1 GeV. The last column shows the threshold-independent neutralino annihilation rate, ΓA.Detector systematic uncertainties have been included in the calculation of the limits. The corresponding numbers withoutincluding uncertainties are shown in parenthesis

mχ[GeV ] annihil. φµ ΓA

channel [×103Km−2yr−1] [s−1]

100 hard 8.9 (6.3) 4.0 (2.9)×1014

soft 133.5 (68.2) 4.3 (2.2)×1016

250 hard 2.1 (1.5) 1.3 (0.9)×1013

soft 6.9 (3.9) 3.8 (2.2)×1014

500 hard 1.5 (1.1) 2.5 (1.8)×1012

soft 2.7 (1.9) 4.4 (3.0)×1013

1000 hard 1.5 (1.2) 6.5 (5.4)×1011

soft 1.8 (1.4) 9.2 (6.8)×1012

3000 hard 1.1 (1.0) 7.5 (6.7)×1010

soft 1.5 (1.3) 1.5 (1.3)×1012

5000 hard 1.1 (1.0) 3.2 (2.8)×1010

soft 1.5 (1.2) 7.6 (6.4)×1011

muon flux at an energy threshold of 1 GeV derived usingequations 2, 3 and 4 are shown in parenthesis in table III.The fluxes have been corrected for the inefficiency intro-duced by using angular cones that include 90% of the sig-nal, so the numbers presented represent the limit on thetotal muon flux for each neutralino model. The thresh-old of 1 GeV has been chosen to be able to compare withpublished limits by other experiments that have similarmuon thresholds (see section VIII below).

B. Evaluation of the limits including systematic

uncertainties

However, the best limits an experiment can set are af-fected by the systematic uncertainties entering the anal-ysis. Including the known theoretical and experimentalsystematic uncertainties in the calculation of a flux limitis not straightforward, and often overlooked in the lit-erature. A precise evaluation of a limit should involvethe incorporation of both the uncertainties in the back-ground counts, σb, and in the effective volume, σV. Anadditional caveat arises since the uncertainty in the effec-tive volume introduces in turn an additional uncertainty

11

Neutralino mass (GeV)

Γ A (

s-1

)

AMANDA-B10

130 d livetime

Hardspectra

Softspectra

10 10

10 11

10 12

10 13

10 14

10 15

10 16

10 17

102

103

FIG. 9: 90% confidence level upper limits on the neutralinoannihilation rate, ΓA, in the center of the Earth as a func-tion of the neutralino mass and for the two extreme anni-hilation channels considered in the analysis. The dashedlines indicate the limits obtained without including sys-tematic uncertainties and correspond to the numbers inparentheses in table III. The symbols indicate the massesused in the analysis. Lines are to guide the eye.

Neutralino mass (GeV)

φ µ (

km-2

yr-1

)

AMANDA-B10

130 d livetimeEthr = 1 GeV

Hard spectra

Soft spectra

10 3

10 4

10 5

10 6

102

103

FIG. 10: 90% confidence level upper limits on the muonflux at the surface of the Earth, φµ, as a function of the neu-tralino mass and for the two extreme annihilation channelsconsidered in the analysis. The dashed lines indicate thelimits obtained without including systematic uncertaintiesand correspond to the numbers in parentheses in table III.The symbols indicate the masses used in the analysis.Lines are to guide the eye.

in the expected number of background events, on topof the 30% uncertainty used in the background neutrinoflux σb. A proper implementation of the systematics inthe calculation of a limit should take this correlation intoaccount.

One approach to incorporate systematic uncertaintiesinto an upper limit has been proposed in Ref. 34. Wehave developed a similar method suited to our specificcase which includes the systematic uncertainty in Veff inthe calculation of N90 used in equation 2. The method isa modified Neyman-type confidence belt construction35.The confidence belt for a desired confidence level β isconstructed in the usual way by integrating the Poissondistribution with mean ntot=nS+nB so as to include a β%probability content. But the number of events for signaland background, nS and nB, are taken themselves to berandom variables obtained from Gaussian distributionswith means equal to the actual number of signal andbackground events observed and widths corresponding tothe systematic uncertainties in signal and background.

Given an experimentally observed number of events,Nexp, the 90% confidence level limit on the number ofsignal events is obtained by simply inverting the calcu-lated N90(ntot) at the corresponding ntot=Nexp value. Inthis way the different uncertainties for signal and back-ground and the correlation between them are includednaturally.

In summary, the inclusion of our present systematicuncertainties in the flux limit calculation yields resultswhich are weakened between ∼10% and ∼40% (practi-cally a factor of 2 for the soft channel of mχ=100 GeV)with respect to those obtained using N90 calculated with-out systematics. The effect is dependent on the WIMPmass, and it reflects the better sensitivity of AMANDAfor higher neutrino energies. Figures 9 and 10 show the90% confidence level limit on the neutralino annihilationrate and the corresponding limit on the resulting muonflux for a muon threshold of 1 GeV for the hard and softannihilation channels considered in the analysis. Thesymbols show the particular neutralino masses used inthe simulation. The lines are to guide the eye and theyshow the limits obtained including systematic uncertain-ties (solid). The dashed lines, included for comparison,show the values obtained using the Neyman construc-tion with the unified ordering scheme without includinguncertainties. Table III summarizes the correspondingnumbers.

C. Effect of neutrino oscillations

To account for neutrino oscillations among the differ-ent flavors, the atmospheric neutrino spectrum shouldbe weighted by a factor W(Eν) which includes the prob-

12

Ethµ = 1 GeV (Super-K: 1.6 GeV)

Neutralino mass (GeV)

φµ

(km

-2 y

r-1)

AMANDA

BAKSANMACROSUPER-K

10 2

10 3

10 4

10 5

10 6

10 102

103

104

FIG. 11: The AMANDA limits on the muon flux from neu-tralino annihilations from figure 10 compared with publishedlimits from MACRO, Baksan and Super-Kamiokande. Thedots represent model predictions from the MSSM, calculatedwith the DarkSUSY package39. The dashed area shows themodels disfavored by direct searches from the DAMA collab-oration45.

ability that a muon neutrino has oscillated into an-other flavor in its way through the Earth to the detec-tor. For the purpose of illustration consider a two-flavoroscillation scenario between νµ and ντ . Then W(Eν)

= 1 − sin2(2θ) sin2(

1.27 ∆m2[eV2] D⊕[km]/Eν [GeV])

,where D⊕ is the diameter of the Earth, θ the mixingangle and ∆m2 the difference of the squares of the flavormasses. Note that the effect depends strongly on the neu-trino energy and it is negligible in the high energy tail ofthe atmospheric spectrum since the oscillation length isthen much larger than the Earth diameter. If we choosesin2(2θ)=1 and ∆m2 = 2.5 × 10−3 eV2 based on the re-sults obtained in Ref. 36, the number of expected atmo-spheric neutrino events is reduced between 5% and 10%,depending on the angular cone considered. This wouldweaken the limits by about the same amount.

The effect of neutrino oscillations on the possibleWIMP signal is model dependent and has been estimatedin Refs. 37 and 38. However the authors reach differentconclusions on the direction of the effect: up to a fac-tor of two in increased muon flux in Ref. 37 and a re-duction of about 25% in Ref. 38 for a neutralino massof 100 GeV. For higher neutralino masses both authorspredict a less pronounced effect, which becomes negligiblefor the higher masses considered in37 (mχ > 300 GeV).We have not included any oscillation effect on the neu-trinos from the WIMP signals considered in this paper.

VIII. COMPARISON WITH OTHER

EXPERIMENTS AND THEORETICAL MODELS

Searches for a neutrino signal from WIMP annihila-tion in the center of the Earth have been performed byMACRO, Baikal, Baksan, and Super-Kamiokande.

In figure 11 the results of Baksan40, MACRO41 andSuper-Kamiokande42 are shown along with the limitsfrom AMANDA obtained in the previous section andtheoretical predictions of the MSSM as a function ofWIMP mass. In order to be able to compare with theother experiments, the Super-Kamiokande limits havebeen scaled by a factor 1/0.9 to represent total flux lim-its, instead of limits based on angular cones including90% of the signal as originally presented in Ref. 42. The90% confidence level muon flux limits for a muon energythreshold of 10 GeV published by the Baikal collabora-tion range between 0.63×104 km−2 yr−1 for a zenith halfcone of 15 and 0.54×104 km−2 yr−1 for a zenith halfcone of 5 (Ref. 43). Since these results are not pre-sented as a function of WIMP mass, and are quoted ata slightly higher muon energy threshold, we have not in-cluded them in the figure but we mention them here forcompleteness.

Each point in the figure represents a flux obtainedwith a particular combination of MSSM parameters, fol-lowing Ref. 44. The sixty four original free parame-ters of the general MSSM have been reduced to sevenby the standard assumptions about the behavior of thetheory at the GUT scale and about the supersymme-try breaking parameters in the s-fermion sector. Theindependent parameters left are the Higgsino mass pa-rameter µ, the ratio of the Higgs vacuum expectationvalues tanβ, the gaugino mass parameter M2, the massmA of the CP-odd Higgs boson and the quantities mo, At

and Ab from the ansatz on the scale of supersymmetrybreaking. These parameters were varied in the follow-ing ranges: -5000≤ µ ≤5000 GeV , -5000≤ M2 ≤5000GeV, 1.2≤ tanβ ≤ 50, mA ≤ 1000 GeV, 100≤ mo ≤ 3000GeV, -3mo ≤ Ab ≤ 3mo and -3mo ≤ At ≤ 3mo. Mod-els based on parameters already excluded by acceleratorlimits are not shown, and the figure is restricted to thosemodels which give cosmologically interesting neutralinorelic densities, 0.025<

∼ Ωχh2 <0.5. A local dark matterdensity of 0.3 GeV/cm3 has been assumed. Theoreticalpredictions for high mass neutralino models lie below thescale of the plot, since in this case the number density ofneutralinos falls down rapidly if the dark matter densityis kept fixed.

A complementary way to search for neutralinos is bymeasuring the nuclear recoil in elastic neutralino-nucleuscollisions on an adequate target material2. Experimentsusing this direct detection technique set limits on theneutralino-nucleon cross section as a function of neu-tralino mass. The same scan over MSSM parameter spaceused to generate the theoretical points in figure 11 canbe used to identify parameter combinations that are ac-cessible by direct searches. There is not, however, a one-

13

to-one correspondence between the results of the directdetection searches and the expected neutrino flux fromthe models probed, so comparisons with the results of in-direct searches have to be performed with care. We haveindicated the models disfavored by the DAMA collabo-ration45 by the dashed area in the figure, which has to betaken as an approximate region in view of the mentioneddifficulties in comparing both types of detection tech-niques. We note that the models that yield high muonfluxes, and that are disfavored by both current resultsfrom direct searches and by the limits shown in the fig-ure, have in common a low value of the H0

2 mass, around92 GeV.

IX. SUMMARY

We have performed a search for a statistically sig-nificant excess of vertically up-going muons with theAMANDA neutrino detector as a signature for neutralinoannihilation in the center of the Earth. Limits on theneutralino annihilation rate have been derived from thenon-observation of a signal excess over the predicted at-mospheric neutrino background. We have included theeffect of the detector systematic uncertainties and thetheoretical uncertainty in the expected number of back-ground events in the derivation of the limits, presentingin this way realistic limit values.

A comparison with the results of MACRO, Super-Kand Baksan, as well as with theoretical expectations fromthe MSSM are presented. AMANDA, with only 130.1days of effective exposure in 1997, has reached a sensitiv-ity in the high neutralino mass (>500 GeV) region com-parable to that achieved by detectors with much longerlive-times.

Acknowledgments

AMANDA is supported by the following agencies: TheU.S. National Science Foundation, the University of Wis-consin Alumni Research Foundation, the U.S. Depart-ment of Energy, the U.S. National Energy Research Sci-entific Computing Center, the Swedish Research Council,the Swedish Polar Research Council, the Knut and AlliceWallenberg Foundation (Sweden) and the German Fed-eral Ministry of Education and Research. D. F. Cowenacknowledges the support of the NSF CAREER program.C.P. de los Heros acknowledges support from the EU4th framework of Training and Mobility of Researches.P. Loaiza was supported by the Swedish STINT pro-gramme. We acknowledge the invaluable support of theAmundsen-Scott South Pole station personnel. We arethankful to I.F.M. Albuquerque and W. Chinowsky fortheir careful reading of the manuscript and valuable com-ments.

∗ now at CERN, CH-1211, Geneve 23, Switzerland.† now at Defense Research Establishment (FOA), S-17290

Stockholm, Sweden.‡ now at Santa Cruz Institute for Particle Physics, Univer-

sity of California - Santa Cruz, Santa Cruz, CA 95064,USA.

§ now at IFAE, 08193 Barcelona, Spain.¶ now at MontaVista Software, 1237 E. Arques Ave., Sun-

nyvale, CA 94085, USA.∗∗ now at The Svedberg Laboratory, S-75121, Uppsala, Swe-

den†† now at Johns Hopkins University, Applied Physics Labo-

ratory, Laurel, MD 20723, USA.‡‡ now at Optical Networks Research, JDS Uniphase, 100

Willowbrook Rd., Freehold, NJ 07728-2879, USA.§§ Corresponding author. E-mail: cph@tsl.uu.se¶¶ now at L-174, Lawrence Livermore National Laboratory,

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